Quasitruncated cuboctahedron

Quasitruncated cuboctahedron
(3D model) (OFF file)
Rank	3
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Quitco
Coxeter diagram	x4/3x3x ()
Elements
Faces	12 squares, 8 hexagons, 6 octagrams
Edges	24+24+24
Vertices	48
Vertex figure	Scalene triangle, edge lengths √2, √3, √2–√2
Measures (edge length 1)
Circumradius	$\frac{\sqrt{13-6\sqrt2}}{2} ≈ 1.06239$
Volume	$2(11-7\sqrt2) ≈ 2.20101$
Dihedral angles	8/3–4: 135°
	8/3–6: $\arccos\left(\frac{\sqrt3}{3}\right) ≈ 54.73561^\circ$
	6–4: $\arccos\left(\frac{\sqrt6}{3}\right) ≈ 35.26439^\circ$
Central density	-1
Number of external pieces	146
Level of complexity	26
Related polytopes
Army	Semi-uniform Girco, edge lengths $\sqrt2-1$ (octagons), $2-\sqrt2$ (ditrigon-rectangle)
Regiment	Quitco
Dual	Great disdyakis dodecahedron
Conjugate	Great rhombicuboctahedron
Convex core	Octahedron
Abstract & topological properties
Flag count	288
Euler characteristic	2
Orientable	Yes
Genus	0
Properties
Symmetry	B3, order 48
Convex	No
Nature	Tame

The quasitruncated cuboctahedron or quitco, also called the great truncated cuboctahedron, is a uniform polyhedron. It consists of 12 squares, 8 hexagons, and 6 octagrams, with one of each type of face meeting per vertex. It can be obtained by quasicantitruncation of the cube or octahedron, or equivalently by quasitruncating the vertices of a cuboctahedron and then adjusting the edge lengths to be all equal.

Vertex coordinates

A quasitruncated cuboctahedron of edge length 1 has vertex coordinates given by all permutations of:

• $\left(±\frac{2\sqrt2-1}{2},\,±\frac{\sqrt2-1}{2},\,±\frac12\right).$

Related polyhedra

o4/3o3o truncations

Name	OBSA	Schläfli symbol	CD diagram	Picture
Cube	cube	{4/3,3}	x4/3o3o ()
Quasitruncated hexahedron	quith	t{4/3,3}	x4/3x3o ()
Cuboctahedron	co	r{3,4/3}	o4/3x3o ()
Truncated octahedron	toe	t{3,4/3}	o4/3x3x ()
Octahedron	oct	{3,4/3}	o4/3o3x ()
Quasirhombicuboctahedron	querco	rr{3,4/3}	x4/3o3x ()
Quasitruncated cuboctahedron	quitco	tr{3,4/3}	x4/3x3x ()
(degenerate, oct+6(4))			o4/3o3ß ()
Icosahedron	ike	s{3,4/3}	o4/3s3s ()

External links

• Bowers, Jonathan. "Polyhedron Category 5: Omnitruncates" (#59).

• Klitzing, Richard. "quitco".

• Wikipedia Contributors. "Great truncated cuboctahedron".
• McCooey, David. "Great Truncated Cuboctahedron"